The value of the parameter a such that the area bounded by `y=a^(2)x^(2)+ax+1,` coordinate axes, and the line x=1 attains its least value is equal to

Options

A. `(1)/(4)` sq. units
B. `-(1)/(2)` sq. units
C. `(3)/(4)` sq. units
D. `-1` sq. units

Answer

C

Explanation

`a^(2)x^(2)+ax+1` is clearly positive for all real values of x, Area under consideration
`A=overset(1)underset(0)int(a^(2)x^(2)+ax +1)dx`
`=(a^(2))/(3)+(a)/(2)+1`
`=(1)/(6)(2a^(2)+3a+6)`
`=(1)/(6)(2(a^(2)+(3)/(2)a+(9)/(16))+6-(18)/(16))`
`=(1)/(6)(2(a+(3)/(4))^(2)+(39)/(8))`, which is clearly minimum for `a=-(3)/(4).`